Hypothesis and Aims: This trial is being conducted as a step toward testing the long-term hypothesis that the costimulation blockade agent abatacept can be added to a standard acute graft-versus-host disease (aGvHD) prophylaxis regimen (which includes a calcineurin inhibitor (CNI) and methotrexate), to improve disease-free survival after unrelated hematopoietic stem cell transplantation (HSCT) for patients with hematologic malignancies. As a phase II study, the overall aim of this trial is to make a preliminary assessment of abatacept's clinical safety and efficacy using short-term outcomes. Thus, this trial is designed to test two hypotheses: